Graph Y = 3(2x + 1): Understanding The Basics
Table of Contents :
To understand the graph of the equation ( y = 3(2x + 1) ), it’s essential to break down the components of the equation, visualize the graph, and analyze its characteristics step by step.
What is the Equation?
The equation ( y = 3(2x + 1) ) is a linear equation in the form of ( y = mx + b ), where:
- ( m ) represents the slope of the line.
- ( b ) is the y-intercept, the point at which the line crosses the y-axis.
Breaking Down the Equation
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Distributing the 3: [ y = 3(2x) + 3(1) = 6x + 3 ] This shows that the slope ( m = 6 ) and the y-intercept ( b = 3 ).
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Intercept Form: The slope-intercept form is now clearly represented: [ y = 6x + 3 ]
Key Features of the Line
Slope (m)
The slope of 6 indicates that for every unit increase in ( x ), the value of ( y ) increases by 6 units. This steep slope suggests the line will rise quickly as you move from left to right across the graph.
Y-Intercept (b)
The y-intercept of 3 means that when ( x = 0 ), ( y = 3 ). This is the point where the line intersects the y-axis.
Creating a Table of Values
To graph the equation ( y = 6x + 3 ), we can create a table of values by choosing different ( x ) values:
<table> <tr> <th>x</th> <th>y = 6x + 3</th> </tr> <tr> <td>-1</td> <td>-3</td> </tr> <tr> <td>0</td> <td>3</td> </tr> <tr> <td>1</td> <td>9</td> </tr> <tr> <td>2</td> <td>15</td> </tr> </table>
Plotting the Points
Using the table we created, we can plot the points on a graph:
- The point ((-1, -3))
- The point ((0, 3))
- The point ((1, 9))
- The point ((2, 15))
Once you have plotted these points, you can draw a straight line through them to represent the equation graphically.
Understanding the Graph
When you visualize the graph of ( y = 6x + 3 ):
- The line will steeply rise due to the high positive slope.
- It will intersect the y-axis at ( (0, 3) ).
- The x-axis intersection can be calculated by setting ( y = 0 ): [ 0 = 6x + 3 \implies x = -\frac{1}{2} ] So, the line also crosses the x-axis at ( (-\frac{1}{2}, 0) ).
The Behavior of the Line
As you move towards positive infinity in the x-direction, the value of ( y ) increases significantly due to the positive slope. Conversely, as ( x ) moves toward negative infinity, ( y ) decreases but remains a linear function, without any curves or bends.
Summary of Key Points
- Equation: ( y = 6x + 3 )
- Slope: 6 (indicating the line rises steeply)
- Y-Intercept: 3 (point where it crosses the y-axis)
- X-Intercept: (-\frac{1}{2}) (point where it crosses the x-axis)
- Type of Function: Linear, with a constant slope.
Importance of Graphing Linear Equations
Graphing linear equations provides a visual representation that can help in understanding relationships between variables. In practical applications, such as economics and science, these relationships can model real-world scenarios like profit, cost, or speed.
With ( y = 6x + 3 ), we can not only analyze how one variable changes concerning another but also predict outcomes based on this model.
Conclusion
Understanding the graph of ( y = 3(2x + 1) ) or, more conveniently expressed as ( y = 6x + 3 ), involves grasping its slope, intercepts, and behavior. Through this exploration, we learn not only the technicalities of linear equations but also the profound implications of graphing and visualization in mathematics and its applications in the real world. Whether for academic purposes or practical applications, mastering this fundamental concept is a stepping stone in the vast landscape of mathematics.